Remarks

The theorems of this chapter can be viewed as generalizations of fixed point theorems. Theorem 17.1 is due to Fan [1969] and is based on a theorem of Browder [1967]. It gives conditions on correspondences μ γ : K → Rm which guarantee the existence of an x ∈ K satisfying μ(x) ∩ γ(x) ≠ ø. Browder proves the theorem for the special case in which μ is a singleton-valued correspondence and γ is the identity correspondence. In this case μ(x) ∩ γ(x) ≠ ø if and only if x is a fixed point of μ. The correspondences are not required to map K into itself; instead, a rather peculiar looking condition is used. In the case studied by Browder, this condition says that μ is either an inward or an outward map. Such conditions were studied by Halpern [1968] and Halpern and Bergman [1968].

Another feature of these theorems, also due more or less to Browder, is the combination of a separating hyperplane argument with a maximization argument. The maximization argument is based on 7.2; which is equivalent to a fixed point argument. Such a form of argument is also used in 18.18 below and is implicit in 21.6 and 21.7.

Fan-Browder Theorem (Fan [1969, Theorem 6])

Let K ⊂ Rm be compact and convex, and let γ,μ : K →→ Rm be upper hemi-continuous with nonempty closed convex values.